Schwartz received his early education at the École Normale Supérieure (now part of the Universities of Paris) and the Faculty of Science, both located in Paris. He received his doctorate in mathematical sciences in Paris, after which he became a professor at the University of Nancy (1945–52). He joined the Faculty of Science (1953–83) and also served as a professor of analysis at the École Polytechnique, Palaiseau (1959–60, 1963–83).

Schwartz was awarded the Fields Medal at the International Congress of Mathematicians in Cambridge, Massachusetts, U.S., in 1950 for his work on the theory of distributions, or generalized functions. For example, prior to the work of Schwartz, physicists concerned with mass distributions used the so-called Dirac delta function, which is 0 when x ≠ 0, is +∞ for x = 0, and has integral equal to 1 over any interval containing 0. It was a useful but limited tool and, from a rigorously mathematical point of view, not a function. In a classic paper, Schwartz showed how to make rigorous sense of a wide variety of objects like this. His ideas about generalized functions later found application in partial differential equations, potential theory, and spectral theory.

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